Ординатура / Офтальмология / Английские материалы / Optics Learning by Computing with Examples using MATLAB_Dieter Moller_2007
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2.5. TWO-BEAM AMPLITUDE DIVIDING INTERFEROMETRY |
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The splitting of amplitude division is different from the splitting discussed for wavefront division. The two waves after wavefront division travel under an angle. After amplitude division, it can be arranged, that both parts travel in parallel in the same direction. The difference between wavefront and amplitude division is related to energy conservation. It is impossible to superimpose two beams in such a way that all the light travels in one direction.
The interference pattern is observed at a faraway screen, and the intensity of the interference pattern is given as I I0[A cos(2πδ/2λ)]2 (see Eq. (2.25)), where δ is the optical path difference. The observation may be done in the focal plane of a lens, making the actual distance between the experiment and observation screen much shorter.
2.5.2 Plane Parallel Plate
The interference on a plane parallel plate is the model for the description of interference on thin films, used in various technologies such as coating lenses or mirrors for use with Xrays.
We consider light incident on a plane parallel plate of glass of thickness D and index of refraction n > 1, and assume n 1 for the media outside the plate. The incident light, shown in Figure 2.9 at (a), is split at the first interface into a reflected and transmitted part (shown at (b)). The transmitted light is reflected and transmitted at the second interface (shown at (c)) and the reflected light is again reflected and transmitted at the first interface (shown at (d)). We use for further consideration only the light reflected from the first interface (shown as (1)) and the light reflected at the second interface, and then transmitted through the first interface (shown as (2)).
FIGURE 2.9 Plane parallel plate at thickness D and refractive index n; (a) incident light; (b) splitting at first interface; (c) splitting at second interface, (d) splitting at first interface again; (e) the two waves, (1) from b and (2) from c superimposed to generate the interference fringes.
98 2. INTERFERENCE
The path difference between the waves (1) and (2) is
δ 2Dn. |
(2.38) |
Both waves (1) and (2) travel in the same direction, and (1) picks up a phase shift of π when reflected at the first interface. For the calculation of the optical path difference we have to multiply 2D by n. In addition, we have to take into account that the reflection on the denser medium, that is, the glass, for wave (1) results in a phase shift of π, equivalent to λ/2. The optical path difference is then
δ 2Dn + λ/2, |
(2.39) |
where λ is the wavelength outside of the plate. One has for the intensity |
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I I0[cos{π(2Dn + λ/2)/λ}]2 |
(2.40) |
or equivalently |
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I I0[cos{π(2Dn)/λ + π/2}]2. |
(2.41) |
We have for constructive interference |
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δ 2Dn + λ/2 0, λ, 2λ, . . . |
(2.42) |
or |
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2Dn λ/2, 3/2λ, 5/2λ, . . . |
(2.43) |
and for destructive interference |
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δ 2Dn + λ/2 λ/2, 3λ/2, 5λ/2, . . . |
(2.44) |
or |
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2Dn λ, 2λ, 3λ, . . . . |
(2.45) |
We see that the appearance of maxima and minima depends on the thickness and the index of refraction of the plane parallel plate. Maxima are obtained for integer numbers of half a wavelength, minima for integer numbers of a wavelength. One may observe the interference pattern on a plane parallel plate by looking at a soap bubble. The film of the bubble is curved, but equally thick over a small area. The colored light we see is produced by individual interference of different wavelengths on the thin film of equal thickness.
The first graph in FileFig 2.9 shows the dependence of the fringes on the thickness of the film for fixed wavelength λ and refractive index n 1.5. The second graph shows the dependence on wavelength for fixed thickness D 0.05 and n 1.5. One observes that there is no interference on the thin film when the wavelength gets too large.
2.5. TWO-BEAM AMPLITUDE DIVIDING INTERFEROMETRY |
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FileFig 2.9 |
(I9PLANS) |
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The intensity of two-beam interference on a plane parallel plate of index n2 in medium with index n1 1. The graphs shown are: (i) Dependence on thickness for fixed wavelength λ 0.0005 and n 1.5; (ii) Dependence on wavelength for fixed thickness D 0.05 and n 1.5.
I9PLANS is only on the CD.
Application 2.9.
1.Modify the formula to I (D) {cos(2πDn/λ + λ/2)}2, for the case that the index of the outside medium is not 1.
2.Consider the following configurations:
a.an air gap between glass media (n 1.5).
b.a water film (n 1.33) on glass, and incident light in medium n 1.
c.a water film on glass and incident light in glass.
Using a graph for fixed D, start counting maxima at any particular maxima on the graph. Read from the graph the difference in D between, for example, 5 maxima. Recalculate the wavelength using the value of n.
3.When the wavelength exceeds the thickness of the plate, the last fringe is observed for a value of λ, depending on D and n. Find the formula.
2.5.2.1 Wedge-Shaped Air Gap
We consider two glass plates in air with one on top of the other. With a thin object, we produce a wedge shaped air gap of small angle α. As we did for the plane parallel plate, we calculate the optical path difference for the two waves
(1) and (2), on the two interfaces of the air gap (Figure 2.10a (side view)). The optical path difference is
δ 2x tan α, |
(2.46) |
where α is the angle of the wedge and x the distance from the point where the plates are touching. The wave reflected at the lower plate picks up a phase shift of π, equivalent to λ/2. For the two waves (1) and (2) one has constructive interference
2x tan α + λ/2 |
0, λ, 2λ, 3λ, . . . , mλ |
(2.47) |
and destructive interference |
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2x tan α + λ/2 |
1/2λ, 3/2λ, 5/2λ, . . . , (m + 1/2)λ. |
(2.48) |
The width D x tan α at the mth maximum is |
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D (m − 1/2)λ/2, |
(2.49) |
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100 2. INTERFERENCE
FIGURE 2.10 (a) Optical diagram for a wedge-shaped gap of air between two dielectric glass plates (microscope slides). The gap angle α is assumed to be small. As x increases, the optical path difference between waves (1) and (2) also increases, resulting in an interference pattern;
(b)schematic of fringe pattern observed with optical plates flat to a fraction of a wavelength λ;
(c)height b at distance xm d and tan α b/d.
where m 1, 2, 3, . . . , and for destructive interference |
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D mλ/2, |
(2.50) |
where m 0, 1, 2, 3, . . . . For the intensity we have |
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I I0[cos{(π2x tan α)/λ + π/2}]2. |
(2.51) |
Interference produced by a wedge-shaped air gap is used to determine the uniformity of a polishing job, schematically shown in Figure 2.10b. A flat plate is used to produce an air gap over a plate, which has been polished. If the maxima and minima are not straight lines, the width of the air gap varies. Deviations of a fraction of a wavelength can be detected.
One can make a simple experiment with two microscope slides and a plastic film. The film is placed at the end of one slide, and a wedge of length approximately 5 cm may be produced with the other microscope slide. Observation of fringes and their corresponding distances makes it possible to determine the thickness of the plastic film.
2.5. TWO-BEAM AMPLITUDE DIVIDING INTERFEROMETRY |
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FileFig 2.10 (I10WEDGES)
The intensity for interference depending on the distance x, of a wedge of α 0.002 rad and λ 0.0005 mm. The distance between the maxima is a constant, given as λ/(2 tan α). We have also plotted the height depending on the length x for angle α, using a scaling factor a. The first fringe is a minimum, as observed for Lloyd’s mirror, because the thickness at origin D 0.
I10WEDGES is only on the CD.
Application 2.10.
1.From the condition of constructive interference (2x tan α + λ/2 2λ, . . . , mλ), show that we have for the difference xm xm+1 λ/(2 tan α), where xm is the x coordinate at the mth fringe.
0, λ,
−xm
2.Since ym/ xm tan α, where ym is the height difference of the plates between fringes, show that we have ym λ/2.
3.Assume we observe M fringes over the length xm d and want to determine the height b of the gap at that point (see Fig. 2.10a. Since b/d tan α (λ/2)/xm, show that we have b (λ/2)M.
4.Recalculation of α: produce a graph with λ .00054 and α .0023. Find the x coordinate at the 23rd fringe and use for the y coordinate y (λ/2)23. Calculate α and compare with input data.
5.Modify the determination of the height at some chosen point xm for a water film (n 1.33) between glass plates (n 1.5).
2.5.2.2 Newton’s Rings
A circular interference pattern may be observed if a spherical surface is placed on a flat surface. The ring pattern is called “Newton’s rings” and may be used to determine the radius of curvature of the spherical surface. An experimental setup is shown in Figure 2.11.
A plane convex lens touches a plane parallel plate and an air gap of width D is formed between the lens and the plate. We call the radius of curvature of the spherical surface R and the radius of the rings of the pattern r (Figure 2.12). One has the relation
R2 r2 + (R − D)2. |
(2.52) |
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After solving a quadratic equation, we have for D(r) |
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D(r) R − (R2 − r2). |
(2.53) |
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The transmitted intensity is |
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I (r) I0{cos(π2D(r)/λ)}2. |
(2.54) |
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102 2. INTERFERENCE
FIGURE 2.11 Experimental setup for the observation of Newton’s rings. Light is made parallel by lens 1 and is reflected by the beam splitter to the lens-plate assembly. Reflected light from surfaces I and II travels to microscope 1 for observation of fringes in reflection. Transmitted light from surfaces I and II travels to microscope 2 for observation of the tranmission fringes.
FIGURE 2.12 Coordinates for the calculation of the optical path difference D between rays from surfaces I and II.
In addition the reflected intensity has the term π/2 because of the reflection on the plane parallel plate of refractive index larger than 1,
I (r) I0{cos(π2D(r)/λ + π/2)}2. |
(2.55) |
2.5. TWO-BEAM AMPLITUDE DIVIDING INTERFEROMETRY |
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FIGURE 2.13 Newton’s rings observed: (a) using transmitted light; (b) using reflected light (from M. Cagnet, M. Francon, and J.C. Thrierr, Atlas of Optical Phenomena, Springer-Verlag, Heidelberg, 1962).
For the reflected intensities, one has for constructive interference
δ D(r) + λ/2 |
0, λ, 2λ, . . . , |
(2.56) |
and for destructive interference |
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δ D(r) + λ/2 |
λ/2, 3λ/2, 5λ/2, . . . . |
(2.57) |
For the transmitted intensities one has similar expressions without the λ/2 term. At the center of the plates, at D 0, one has from Eq. (2.55) for reflected light zero intensity; in other words, we should observe a dark spot, shown in Figure 2.13a. For transmitted light, Eq. (2.54) predicts a bright spot, shown in Figure 2.13b.
FileFig 2.11 (I11NEWTONS)
Intensity for Newton’s rings in transmission and reflection depending on the radius r around the center, for λ 0.0005 mm and R 2000 mm.
I11NEWTONS is only on the CD.
Application 2.11. Recalculation of R:
1.Show that we have for the mth ring (fringe) a height in the air gap of mλ/2.
2.Show that we then have to use (mλ/2 − R)2 R2 − rm2 for the calculation of R, assuming that we have read rm from the graph.
2.5.3Michelson Interferometer and Heidinger and Fizeau Fringes
2.5.3.1 Michelson Interferometer and Normal Incidence
In 1880, Albert Michelson used the interferometer, named after him, for his famous experiments to show that there is no ether. Today most infrared spec-
104 2. INTERFERENCE
FIGURE 2.14 The Michelson interferometer with arms of unequal lenght x1 and x2 in the same medium.
trometers use the Michelson interferometer to obtain an interferogram and the application of the Fourier transform produces the desired spectrum. Chapter 9 discusses Fourier transformation and spectroscopy. In Figure 2.14 we show a schematic of a Michelson interferometer. The amplitude of the incident beam is partly reflected under an angle of 90◦ toward mirror M1, and partly transmitted in the direction to mirror M2. The beam splitter may be a plane parallel plate, and the reflection and transmission properties of such a plate are discussed below, including all multiple reflections. Here we use an idealized beamsplitter and assume that 50% of the incident light will be reflected, that 50% will be transmitted, and no phase shift will be introduced.
The reflected wave travels to mirror M1, where it is reflected and travels back to the beam splitter for a “second splitting.” It has traveled the distance 2x1 between mirror and beam splitter. We call the transmitted part (1), and note that the reflected part travels back to the source. Similarly, the transmitted wave travels to M2, is there reflected, and travels to the beam splitter for a second splitting. It has traveled the distance 2x2 and we call the reflected part (2); the transmitted part travels back to the source. Parts (1) and (2) are superimposed and travel to the detector. If the distances 2x1 and 2x2 are not the same, we have an optical path difference between (1) and (2) of
δ 2D 2(x2 − x1), |
(2.58) |
where we assume that x1 > x1. Constructive interference is obtained for |
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δ mλ |
(2.59) |
2.5. TWO-BEAM AMPLITUDE DIVIDING INTERFEROMETRY |
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and destructive interference for |
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δ m + 21 λ, |
(2.60) |
where m 0, 1, 2, 3. |
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The intensity of interference is obtained as |
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I 4A2 cos2(π2D/λ). |
(2.61) |
The Michelson interferometer was originally designed to perform exact length measurements. Some time ago, it was used for a now outdated procedure to define the length of the meter by using 86Kr emission.
The first graph in FileFig 2.12 shows the fringes depending on thickness D. The second graph shows the fringes depending on wavelength λ.
FileFig 2.12 |
(I2MICHDLS) |
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Intensity of the Michelson interferometer depending on the displacement D of one mirror for wavelength λ .0005 mm, and for dependence on λ for D .003.
I12MICHDLS
Michelson Interferometer
Beam splitter is assumed to be a plane parallel plate. Fringe pattern depending on D for wavelength λ .0005, and depending on wavelength λ for D .003. The angle θ 0. All lengths in mm.
1. Dependence on D.
θ : 0 |
λ; .0005 |
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D : 0.027, .02701 . . . .0325 |
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I 1(D) : |
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cos |
2 · π · D · cos(θ) |
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λ |
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106 2. INTERFERENCE
2. Dependence on λ
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λ : .0004, .000401 . . . .0008 |
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D : .003 |
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I 2(λ) : |
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cos |
2 · π · D · (θ) |
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λ |
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Application 2.12.
1.Resolution depending on displacement D. Add to the graph of the intensity depending on D a graph with a second wavelength λλ λ+ λ; for example, λλ 0.00052. Observe that the separation of the fringes gets larger for larger m. For the mth fringe we have the path differences mλλ and mλ. When this difference is λ/2, we call the two fringes resolved and have mλλ−mλ λ/2, m λ λ/2, or λ/ λ 2m. Compare the formula λ/λ 2m with values read from the graph for choice of λλ.
2.Add to the graph of the intensity depending on λ a second graph with different D value. The graph shows the change in phase for one wavelength when D is changed. Choose D1 such that maxima change to minima and D2 that mimima change to the next maxima. Read from the graph the numerical values and compare with the formula for constructive and destructive interference.
2.5.3.2Michelson Interferometer, Nonnormal Incidence, Heidinger, and Fizeau Fringes
If the light from the point source fills a cone with opening angle θ, the distance x2 − x1 depends on the angle θ and a ring pattern will result in the plane of the observation screen. For the mathematical treatment we fold one beam of the Michelson interferometer over to the other beam as shown in Figure 2.15a. To calculate the path difference of the two beams (1) and (2), we use Figure 2.15b and calculate the path difference using Figure 2.15c. We have for the distances [ab] [bc] D/ cos θ, and for the distance [ac] [(2D/ cos θ)(sin θ)] 2D tan θ. The optical path difference δ is then
δ 2[bc] − [ac] sin θ 2D/(cos θ) − 2D tan θ sin θ 2D cos θ. (2.62)